350 
of the hodograph, or of the orbit (without violating the re- 
ceived meaning of the term). 
Whatever the value of this numerical eccentricity may be, 
the constant area of the parallelogram under the vectors of 
position and velocity may always be treated as the sum or 
difference of two other parallelograms, of which one is equal 
to the rectangle under the constant radius of the hodographic 
circle and the varying radius vector of the orbit, while the 
other is equal to the parallelogram under the vectors of posi- 
tion and eccentricity ; and hence it is not difficult to infer, 
that the length of the vector of position, or of the radius vec- 
tor of the orbit, varies in a constant ratio, expressed by the 
numerical eccentricity, to the perpendicular let fall from its 
extremity, that is, from the position of the body, on a constant 
straight line or directrix, which is situated in the plane of the 
orbit, and is parallel to the hodographie vector of eccentricity. 
The orbit, therefore, whether it be closed or not, is always 
(with the law of the inverse square) a conic section, having 
the centre of force for a focus—a theorem which has, indeed, 
been known since the time of Newton, but has not, perhaps, 
been proved before from principles so very elementary.* 
Conceive a diameter of the hodograph to be drawn in a 
direction perpendicular to the vector of excentricity ; the ex- 
tremities of this diameter correspond to the extremities of that 
chord of the orbit which is perpendicular to the shortest radius 
* The hodograph of the earth’s annual motion may be considered to be ex- 
hibited to observation in astronomy, as the curve of aberration of a star; and 
it is known that this aberratic curve is a circle, notwithstanding the eccen- 
tricity of the earth’s orbit ; but the author is not aware that this circularity 
of the aberratic curve (for a star near the pole of the ecliptic) has ever been 
shewn before to be a consequence of the law of the inverse square, except by the 
help of the properties of the elliptic orbit; whereas the spirit of the present 
communication is to derive that orbit from the circle, and to regard that circle 
itself as a sort of geometrical picture of Newton’s law, instead of being only 
one of many corollaries from the laws of Kepler. 
